Optimal. Leaf size=54 \[ 2 A \sqrt {a+b x}-2 \sqrt {a} A \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+\frac {2 B (a+b x)^{3/2}}{3 b} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {80, 50, 63, 208} \begin {gather*} 2 A \sqrt {a+b x}-2 \sqrt {a} A \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+\frac {2 B (a+b x)^{3/2}}{3 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 80
Rule 208
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (A+B x)}{x} \, dx &=\frac {2 B (a+b x)^{3/2}}{3 b}+A \int \frac {\sqrt {a+b x}}{x} \, dx\\ &=2 A \sqrt {a+b x}+\frac {2 B (a+b x)^{3/2}}{3 b}+(a A) \int \frac {1}{x \sqrt {a+b x}} \, dx\\ &=2 A \sqrt {a+b x}+\frac {2 B (a+b x)^{3/2}}{3 b}+\frac {(2 a A) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=2 A \sqrt {a+b x}+\frac {2 B (a+b x)^{3/2}}{3 b}-2 \sqrt {a} A \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 55, normalized size = 1.02 \begin {gather*} A \left (2 \sqrt {a+b x}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )+\frac {2 B (a+b x)^{3/2}}{3 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.03, size = 57, normalized size = 1.06 \begin {gather*} \frac {2 \left (3 A b \sqrt {a+b x}+B (a+b x)^{3/2}\right )}{3 b}-2 \sqrt {a} A \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.20, size = 111, normalized size = 2.06 \begin {gather*} \left [\frac {3 \, A \sqrt {a} b \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (B b x + B a + 3 \, A b\right )} \sqrt {b x + a}}{3 \, b}, \frac {2 \, {\left (3 \, A \sqrt {-a} b \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (B b x + B a + 3 \, A b\right )} \sqrt {b x + a}\right )}}{3 \, b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.34, size = 55, normalized size = 1.02 \begin {gather*} \frac {2 \, A a \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {2 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} B b^{2} + 3 \, \sqrt {b x + a} A b^{3}\right )}}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 46, normalized size = 0.85 \begin {gather*} \frac {-2 A \sqrt {a}\, b \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+2 \sqrt {b x +a}\, A b +\frac {2 \left (b x +a \right )^{\frac {3}{2}} B}{3}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.96, size = 60, normalized size = 1.11 \begin {gather*} A \sqrt {a} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + \frac {2 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} B + 3 \, \sqrt {b x + a} A b\right )}}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 45, normalized size = 0.83 \begin {gather*} 2\,A\,\sqrt {a+b\,x}+\frac {2\,B\,{\left (a+b\,x\right )}^{3/2}}{3\,b}+A\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.78, size = 54, normalized size = 1.00 \begin {gather*} \frac {2 A a \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 A \sqrt {a + b x} + \frac {2 B \left (a + b x\right )^{\frac {3}{2}}}{3 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________